Possibility of massless Dirac fermions in an Aubry–André–Harper potential

Since Anderson’s groundbreaking work,¹ the investigation into quantum localization has remained a cornerstone of condensed matter physics. Despite extensive research into localization phenomena over decades, numerous intriguing questions persist in this domain. The Anderson model and the Aubry–André–Harper (AAH) incommensurate potential model continue to captivate physicists’ attention. While the Anderson model introduces disorder through random amplitudes in lattice on-site positions, the AAH model^2–4 stands out by imposing disorder derived from a potential function lacking commensurability with the lattice.

Localization effects in the AAH model stem from the intricate interplay between destructive interference during multiple scattering processes of a particle traversing an incommensurate medium and the spectral characteristics of its Schrödinger equation.⁵ Recent years have seen a surge in interest in this area, primarily fueled by innovative experimental studies using optical lattices and cold atomic gases.^6–9 

In particular, the AAH model in optical lattices has posed an intriguing challenge, shedding light on the elusive metal–insulator transition that occurs with variations in the periodic potential’s intensity.^10–13 Researchers have explored delocalization phenomena within the AAH potential, notably utilizing ultracold bosons.^14,15 Recent studies have contributed to understanding the interaction range in both Anderson and AAH systems.¹¹ Xu et al.¹³ revealed a Griffiths-like regime in interacting AAH models. Investigations into the nonlinear response of interacting bosons in quasiperiodic potentials have also provided valuable insights.⁶ In addition, the concept of many-body localization within incommensurate systems has been examined by Huang et al.⁹ A recent report on hopping tunneling through a slowly varying potential is also noteworthy.¹⁶ 

It is important to note that experimental demonstration of an AAH transition in real materials remains a significant challenge. While the detection of an AAH transition has been accomplished through optical measurement techniques using ultracold atoms,² solid-state superlattices have emerged as a promising platform for demonstrating this transition in real crystals. Solid-state superlattices involve the alternating growth of two different materials, with one acting as a potential barrier and the other as a quantum well.^17,18 These structures have garnered considerable attention due to their potential applications in novel devices.^19–22 Recent studies have suggested the existence of phonon Anderson localization in Si/Ge aperiodic superlattices, evident through their thermal conductivity behaviors.^19–21 

Moreover, the captivating Klein paradox, an extraordinary phenomenon emerging from relativistic physics and featuring massless Dirac fermions, has been a focal point in recent scientific discourse. In the groundbreaking theoretical investigation by Katsnelson et al.,²³ the researchers showcased an impeccable transmission for massless Dirac fermions approaching a one-dimensional (1D) potential barrier within a solid-state apparatus. This discovery stands as a solid-state analog^24–30 to the manifestation of the Klein paradox, an intriguing relativistic tunneling effect initially postulated by Klein in the context of 1D potentials.³¹ The implication is that a complete confinement of massless Dirac fermions within a solid-state crystal through a 1D potential barrier becomes unattainable.

Theoretical propositions have surfaced regarding the containment of massless Dirac fermions in graphene, suggesting the utilization of circular graphene p–n junctions.^32–36 Experimental endeavors, employing scanning tunneling microscopy (STM), have successfully revealed that massless Dirac fermions can undergo a temporary entrapment within circular p–n junctions. The wave functions of these charge carriers are confined, adopting quantified modes through continuous reflection near the interface of closed-circle potential barriers.^37–40 These outcomes not only confirm the anomalous anisotropic transmission of massless Dirac fermions at p–n junction boundaries, known as Klein tunneling in graphene,²³ but also provide insights at the atomic scale.

With these considerations in mind, our objective is to evaluate the applied bias effects on an incommensurate potential (Fig. 1). To date, there have been no measurements of carrier transport using incommensurate potentials in real materials. A pivotal aspect of this research area revolves around comprehending electric field effects on quasiperiodic systems. In our tight-binding model, we combine Anderson-type localization, produced by an external bias, and AAH-type localization, given by the incommensurate potential. Intriguingly, we also uncover a novel phenomenon: the presence of massless Dirac fermions in an AAH system at certain values of an applied electric field.

In addition, we have found that fermion eigenvectors reflect the specific characteristics of the quasiperiodic eigenstates. They can be extended or localized in the AAH potential. We will show the possibility of detecting such massless carriers by means of Klein tunneling through a solid-state device. The existence of Dirac carriers in an incommensurate potential can also be employed to demonstrate the AAH metal–insulator transition in a solid-state material, which has not yet been detected. Our overarching goal is to underscore the critical role of field effects in advancing our understanding of experiments involving incommensurate quantum systems. Through these insights, we aspire to make a substantial contribution to the fundamental understanding of the mechanisms governing the massless electron behavior in solid-state superlattices and real crystals.

It is vital to emphasize that the particular form of the on-site potential V_n in Eq. (1) entirely defines the physical model under examination. For instance, a periodic V_n corresponds to the well-known Bloch case, while a random V_n corresponds to the Anderson localization model.¹ In periodic potentials, all states extend indefinitely, whereas random potentials V_n lead to entirely localized states.^10,11 The incommensurate scenario lies between periodic and random cases. In an AAH system, V_n is periodic, but the period is not a rational multiple of the underlying lattice.^12–15 

Consistent with prior work,¹² we have selected $α=2/(5−1)=0.61803$⁠, characterizing quasiperiodicity in certain quasicrystals in the incommensurate case¹² (see Figs. 1 and 2). When α is a real number, strictly speaking, the potential is (almost) periodic given that n is an integer and N is a finite number. In this work, we refer to the potential as periodic when α = 0.6, and as quasiperiodic, or Aubry–André–Harper (AAH), or incommensurate when α = 0.618 032. Figures 1 and 2 illustrate both periodic and quasiperiodic potentials. Our findings are corroborated by averaging across various ϕ values.

Figures 4 and 5 portray the density of states and the Lyapunov exponent as a function of energy for an applied bias of V = 0. The parameters λ are set to λ = 0.5 and λ = 2.0, and the energy reference point is placed at the band center. Across various energy strengths, the potential structure described by Eq. (2) yields three energy bands within the lattice. The eigenvalues cluster in the upper, central, and lower regions, generating the peaks depicted in Figs. 4 and 5 within the density of states. In these eigenvalue clusters, the Lyapunov exponent reaches its minimum values, indicating extended states. Remarkably, no significant disparities are evident between the density of states and Lyapunov exponent plots.

Figure 6 illustrates the dispersion average σ_p as a function of potential strength λ, following Eq. (10). We considered different applied biases, V = 0, V = 0.1, and V = 0.2. At V = 0, as λ reaches its critical point (λ = 2), an AAH metal–insulator transition occurs within the incommensurate model. For λ > 2, σ_p tends toward zero, indicating highly localized eigenstates. Conversely, at λ < 2, σ_p remains nearly equal to 1, signifying predominantly extended eigenvectors. Figure 6 emphasizes the feasibility of detecting the AAH transition using the proposed σ_p indicator.

We would like to emphasize that the AAH potential exhibits the characteristic of being self-dual, with the Hamiltonian maintaining the same form in both momentum and space representations. Consequently, our tight-binding model possesses a self-duality point at λ = 2, delineating the boundary between the localized phase, where all states are confined, and the delocalized phase, where all states extend. At this transition point, the energy spectrum takes on a fractal nature. Nevertheless, the introduction of an electric field disrupts the system’s self-duality. We posit that the fractal behavior of the critical states at the transition point diminishes as F is increased.

Furthermore, the applied bias induces Wannier–Stark localization in the eigenfunctions. Consequently, the initially extended eigenfunctions from the delocalized phase transition into localized states, with the states in the already localized phase exhibiting an even greater confinement. This effect is vividly illustrated in Fig. 6, which demonstrates a decrease in σ_p as V is increased. For cases where λ > 2, the eigenstates show a strong localization when V > 0. In addition, at λ < 2, observable Wannier–Stark localization effects are evident in the eigenvectors.

In Fig. 6, σ_p exhibits a sharp behavior in the absence of an external electric field. Conversely, the averaged dispersion becomes smoother when V > 0. We attribute the sharp behavior to the presence of a self-duality point in the model at λ = 2 and V = 0. The self-duality of the model is disrupted by the introduction of the electric field.

Figure 7 depicts the average position, ⟨n⟩, within the lattice as a function of position n for λ = 0.5 and λ = 2.5. At λ = 0.5, the ⟨n⟩ values are concentrated around the midpoint of the lattice, approximately ⟨n⟩ ∼ 250 for a lattice of size N = 500. This outcome aligns with the prevalence of extended states at λ = 0.5, where the probability of finding a carrier is uniform across all sites, resulting in an average position of approximately ⟨n⟩ ∼ 250, the central site of the tight-binding chain.

Conversely, at λ = 2.5, the ⟨n⟩ values are dispersed across all lattice sites, reflecting a strong localization of most eigenstates. Unlike the λ = 0.5 case, there is no privileged value for ⟨n⟩ at λ = 2. For each distinct eigenvector of the system at λ = 2.5, a different ⟨n⟩ value is obtained. This provides a clear visualization of the AAH transition through the ⟨n⟩ indicator.

Now, considering the effects of an electric field (V > 0) on the system, Fig. 6 illustrates the dispersion average σ_p for small applied biases. The metal–insulator transition at λ_c shifts to a lower λ value if V > 0. At V = 0, the strongly localized regime (σ_p ∼ 0) is reached at λ = 2.5. However, with V = 0.2, Fig. 6 shows that σ_p ∼ 0 at λ = 2.1. Both Anderson and AAH localization models are present in our quantum system when F > 0, resulting in an increased localization of carrier wave functions within the incommensurate lattice.

For deeper insights into this phenomenon, Fig. 8 displays the density of states and the Lyapunov exponent vs energy at λ = 2.0 and V = 1.55. Three peaks in the density of states correspond to the three energy bands observed in the V = 0 case. However, the energy gaps of the AAH potential vanish at V = 1.55. The arrows in Fig. 8 indicate the positions of the energy bandgaps observed when the applied bias is null. Notably, the Lyapunov exponent does not vanish along the energy axis, indicating a localization of all system eigenvectors at λ = 2.0 and V = 1.55.

At λ = 2.0, Fig. 9 displays the band structure of our AAH lattice for two different electric fields, V = 0 and V = 1.55. Band crossings in the energy bands occur at the critical value V_c = 1.55. For applied biases lower than V_c, Dirac cones in the energy bands are not observed. The arrows in Fig. 9, plotted also in Fig. 8, indicate the positions of the band crossings. At V = 0, the eigenvalues are grouped into clusters, as depicted in Fig. 9.

To illuminate this phenomenon further, Fig. 10 illustrates the formation of the upper Dirac cone of Fig. 9. Three different electric fields, V = 0.00, V = 1.00, and V = 1.55, are considered. The arrow indicates the band crossing, and the two straight lines denote the position of the Dirac cone. The central and upper energy bands do not converge if V < V_c.

To explore if this effect is specific to the chosen potential value (λ = 2.0), Figs. 11 and 12 demonstrate the results for a higher potential (λ = 2.5). In addition, Figs. 13 and 14 exhibit the calculations for a lower λ value (λ = 1.0). In Figs. 11–14, the arrows mark the positions of the Dirac crossings. Notably, extended states are observed at the band center in Fig. 13, where the Lyapunov exponent reaches 0 at E = 0. The calculated critical values for the applied bias V_c vary with changes in λ. At λ = 2.5, V_c equals V_c = 1.65, and V_c = 0.95 for the λ = 1.0 case. Figure 15 depicts the calculated critical value for the applied bias V_c vs λ. As the potential strength λ increases in our AAH model, the obtained V_c values also increase.

To verify the robustness of the results under variations in key parameters of the model, such as V and α, we calculated V_c for different α′ values, N = 500. Specifically, at α′ = α + 0.1 and λ = 2, we obtained V_c = 1.58. In addition, for α′ = α − 0.1 and λ = 2, V_c = 1.52 was found.

The Hamiltonian acts on states represented by the two-component spinor $Ψ=[ψa,ψb]T$⁠, where ψ_a and ψ_b correspond to the envelope functions associated with the probability amplitudes. The low-dimensional spectrum of free carriers is E = ±v_Fk, with k denoting the wave vectors around the cones at the Brillouin zones. The signs + and − refer to the electron and hole bands, respectively. It is noteworthy that the wave function of these massless particles can be either extended or localized based on the λ value. The generated fermions mirror the localization characteristics of the incommensurate potential eigenfunctions.

To elucidate this aspect, Fig. 16 displays the plots of two eigenvectors for both cases. These eigenvectors are accompanied by Gaussian functions computed using Eq. (8). It becomes evident that at λ = 1.5, the state in the incommensurate model is localized, represented by slender Gaussian functions with minimal σ dispersion values. Conversely, the AAH model demonstrates an extended state at λ = 0.5, illustrated by broader Gaussian curves.

It is important to note that, through the adjustment of the applied bias, we have the capability to intentionally manipulate the band structure, achieving a true crossing of two energy bands known as a Dirac point. As illustrated in Fig. 10, the Dirac cone, marked by an arrow, is achieved through the crossing of the second and third bands of the system at λ = 2. In the proximity of such crossings, the dynamics can be effectively described by a one-dimensional Dirac equation.⁴⁵ 

In a Dirac cone, the states of electrons and holes are interconnected, exhibiting features similar to the charge-conjugation symmetry observed in quantum electrodynamics. This symmetry arises from the inherent potential symmetry in the system, as particles need to be described by two-component wave functions. These wave functions are crucial for determining the relative contributions of up and down sublattices in the composition of effective particles. The two-component description closely parallels the use of spinor wave functions in quantum electrodynamics. In our condensed-matter system, the carriers effectively emulate Dirac fermions as observed in quantum electrodynamics.²³ 

In Fig. 10, it is evident that when the electric field is below its critical value (V = 1.0), an energy gap exists between both up- and down-cones. In this scenario, the system exhibits massive fermions rather than massless fermions in the potential. To generate massless fermions in the system, a necessary condition is that the applied bias value reaches V_c = 1.55. It is worth noting that there are materials where external fields are not required to have massless fermions in the quantum system. For instance, graphene possesses energy bands with Dirac cones inherent to its atomic structure, and these cones are naturally generated.

Figure 17 exhibits the positions of the Dirac cones for upper and lower massless fermions concerning the AAH potential strength. As λ increases, the energy gap between the fermions also increases. Furthermore, Fig. 18 portrays the Fermi velocity v_F of the massless carriers against λ. Different V_c values have been utilized to compute v_F for each potential strength value. The observation indicates that v_F escalates with increasing λ.

In conclusion, we posit that the existence of massless particles in an AAH potential can be experimentally detected. A potential experimental setup involves measuring the tunneling current^46–48 through an incommensurate potential when an AAH device is inserted between two distinct electrodes. In principle, if the incommensurate potential is in its insulating phase, where all states are localized, carrier transport through the potential will be impractical. Consequently, the expectation of a finite value for the tunneling current through the device under such conditions may not be feasible.

However, considering the relativistic nature of the band structure within our quantum system, the presence of Dirac fermions allows for their transmission through high potential barriers via the Klein tunneling effect. Therefore, even if the system is in its insulating phase, an anticipated current resulting from Klein tunneling may be expected, contingent upon the generation of massless particles within the system.

Our approach involves utilizing a critical value for the external applied bias V_c to facilitate the creation of massless Dirac fermions at the energy crossings within the band structure. Consequently, even though the system is in an insulating phase, a Klein tunneling current can be generated due to the existence of these massless particles. Such an experimental outcome not only holds the potential to serve as evidence of massless carriers but could also represent a groundbreaking demonstration of an AAH metal–insulator transition in a solid-state device, given that this particular transition has yet to be realized experimentally.

In recent years, various platforms have been employed to investigate the AAH potential. The AAH model has been implemented in photonic crystals, cold-atom systems, and superconducting circuits.⁴⁹ Cold atoms, being ionizable and susceptible to electric fields, are particularly noteworthy in this context.

The emergence of Dirac cones in an AAH system holds promise for potential applications in new experiments and devices. For instance, when the system is in the localized phase and V < V_c, no tunneling current through the device is expected. However, once the V value reaches V_c, Klein tunneling through the AAH potential becomes feasible, facilitated by the existence of Dirac cones in the energy bands. This opens the possibility of implementing a new resonant tunneling-type device.

From both an experimental and theoretical standpoints, exploring many-body effects in the AAH model would be intriguing. Recent investigations using optical lattices and cold atomic gases have examined many-body localization effects in an AAH system.⁹ Depending on the strength of interactions, researchers identified three distinct regimes in the quantum system. In our scenario, we speculate that if many-body interactions play a significant role, the formation of Dirac cones and the critical applied bias values could also be influenced by many-body effects.

In summary, our investigation has provided a comprehensive analysis of field effects on incommensurate quantum systems, focusing on a one-dimensional tight-binding model embedded between two electric contacts and subject to an Aubry–André–Harper potential influenced by an external electric field. Our study amalgamated Anderson and Aubry–André–Harper localization effects on carrier wave functions within our model, further elucidating our findings through the computation of the Lyapunov exponent.

Several intriguing phenomena have been unraveled through our exploration of the Aubry–André–Harper potential. Initially, our observations highlighted a metal–insulator transition occurring at the critical potential strength λ_c, leading to a profound localization of carrier eigenvectors within the lattice. Notably, we discovered that this critical potential strength, λ_c, is contingent upon the applied bias value V, diminishing as the external electric field F increases.

In addition, we unveiled the potential to create massless Dirac fermions within the system by carefully adjusting the applied external bias to specific critical values. Consequently, we achieved Dirac crossings within the band structure at defined energies, guiding the dynamics of fermion carriers through a Dirac Hamiltonian characterized by a Fermi velocity, v_F. Intriguingly, our exploration showcased that the wave functions of these massless particles can oscillate between extension and localization contingent upon the strength of the Aubry–André–Harper potential, λ. This intriguing property reflects the localization traits inherent in the Aubry–André–Harper eigenstates.

Finally, our study emphasizes the viability of detecting these massless particles through experimental setups. The potential for such an experimental verification opens doors to probing the elusive Aubry–André–Harper metal–insulator transition within tangible materials. Our research, shedding light on carrier behaviors in incommensurate systems, holds significant promise for the broader research domain.

The authors have no conflicts to disclose.

M. Cruz-Méndez: Investigation (lead); Methodology (lead); Software (lead). H. Cruz: Conceptualization (lead); Supervision (lead); Writing – original draft (lead); Writing – review & editing (lead).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.